The curve that survives censoring

The estimator that uses censored subjects properly recovers the truth. Across fifteen hundred simulated studies the true survival at two years is 0.497 and Kaplan–Meier gives 0.499, where the two naive readings give 0.189 and 0.370.

The construction #

The estimator answers a question that avoids the problem entirely, and the trick is in the question rather than the arithmetic.

Instead of asking “what fraction survive to two years”, which requires knowing what happened to everyone up to two years, ask a sequence of much smaller questions: at each moment an event occurs, what fraction of those still at risk survived it?

At a time when there are 84 subjects still being observed and 3 have the event, the fraction surviving that instant is 1 − 3/84. Multiply those fractions together across all the event times up to two years, and the product is the probability of surviving all of them:

$$\hat{S}(t) = \prod_{t_i \le t} \left(1 - \frac{d_i}{n_i}\right)$$

Each factor is estimated from the subjects available at that moment, which is why censoring stops being a problem. A subject censored at eighteen months appears in the denominator of every factor before eighteen months and in none after — contributing exactly the information they carry, and nothing else.

The denominator is the whole idea #

The risk set is where censoring is handled, and it is worth looking at directly.

It starts at the full cohort and falls for two different reasons: a subject has the event, or a subject leaves. Both remove them from every later denominator, and the estimator does not distinguish — what matters is whether they were present to be observed, not why they stopped being present.

The distinction that does matter is in the numerator. An event contributes to d and pulls the curve down; a censoring does not. So a censored subject reduces the precision of every later estimate without biasing it, which is exactly the right behaviour and is what neither naive reading achieves.

The site’s gate checks the structural property directly — the risk set never grows — because a bug in the bookkeeping would produce a plausible curve, and an estimator whose denominator went up somewhere would be counting a subject twice.

What one step is worth, and what a censored subject does #

Two small computations make the construction concrete before the argument for it.

The step. With 84 at risk and 3 events, the factor is $1 - 3/84 = 0.9643$ — the curve drops by 3.6% at that instant and by nothing at all between events. A curve built this way is a staircase whose treads are the intervals with no events in them, and its shape carries no information: only the risers do.

The uncertainty. Greenwood’s formula sums $d_i / [n_i(n_i - d_i)]$ over the steps and multiplies by $\hat S(t)^2$. That single step contributes $3/(84 \times 81) = 0.00044$, so twenty steps of comparable size give a summed term of about 0.0088 and, at a survival of 0.8, a standard error of $0.8\sqrt{0.0088}$, which is 0.075.

And the censored subject. Somebody censored at eighteen months sits in $n_i$ for every step up to eighteen months and in none afterwards. They are never counted as a survivor and never as a death; they contribute their information for exactly as long as they were observed and then stop.

That is the whole of what the estimator does with censoring, and it is why nothing in the construction requires knowing what happened to them afterwards.

Why a product and not a proportion #

A question worth answering, because the product form looks like a complication over simply counting.

With no censoring, the two agree exactly. The product telescopes into the proportion still event-free, and nothing is gained.

With censoring, the proportion cannot be computed, because the numerator is unknown — the number who survive to two years includes subjects whose status at two years was never observed. The product avoids ever needing that number: it only ever needs, at each event time, how many were at risk and how many had events, and both are observed exactly.

That is the structural reason the estimator exists. It decomposes an unobservable quantity into a product of observable ones, and the decomposition is what censoring makes necessary.

The same manoeuvre appears elsewhere in the subject whenever a joint quantity is inaccessible and its conditional pieces are not, and it is worth recognising as a technique rather than as a formula specific to survival.

The uncertainty #

The curve is an estimate and needs an interval, and the standard error has a closed form — Greenwood’s:

$$\text{se}(\hat{S}(t)) = \hat{S}(t)\sqrt{\sum_{t_i \le t}\frac{d_i}{n_i(n_i - d_i)}}$$

The sum accumulates a term at each event time, and the terms grow as the risk set shrinks — a death among twelve remaining subjects contributes far more uncertainty than one among a hundred and twenty.

So the interval widens to the right, steeply at the end where few subjects remain. That shape is the honest representation of what the study knows: a great deal about early survival and very little about late — though whether the interval is also in the right place at the end of the curve is a separate question from whether it is wide enough.

The site’s gate uses this as its check on the estimate, and the form of the check is worth noting. Rather than requiring the estimate to be close to the truth by some chosen tolerance, it requires the estimate to be within four of its own standard errors of the truth at each of several times. That is a stronger and more meaningful test: it asks whether the estimator’s stated uncertainty is honest, not merely whether the point estimate is good.

An estimator that was accurate but understated its own error would pass a tolerance check and fail this one.

What it takes to break it #

The estimator recovers the truth at every censoring level tested, and the sweep is worth reading carefully because it shows where the robustness comes from and where it stops.

At 37% censoring: Kaplan–Meier 0.498, complete-case 0.221. At 46%: 0.498 and 0.186. At 58%: 0.498 and 0.133. At 71%: 0.499 and 0.072.

The proper estimator does not degrade at all across that range. The naive one degrades monotonically, and by the end is off by more than four times the true value’s distance from zero.

What does break it is the assumption a censored subject’s third state rests on: censoring independent of the event. That is not a robustness question and no amount of data addresses it, because dependent and independent censoring produce identically-shaped datasets. Kaplan–Meier is exactly as wrong as the dependence is strong, and it gives no indication.

So the honest summary of the estimator’s properties: completely robust to how much censoring there is, and completely dependent on why.

What the curve does not support #

Three limits, in the order they cause trouble.

Nothing past the last observation. The curve is undefined beyond the largest observed time, and if that time is a censoring rather than an event, the curve simply stops above zero. Software will sometimes extend it flat, which is a choice rather than an estimate, and reading survival at a time past the follow-up period is reading a convention.

No mean. As the argument for censoring as information noted, a mean survival time requires the whole tail, and the tail is what was not observed. Any reported mean is assuming something about the unobserved region — usually that the curve drops to zero at the last observation, which understates the mean by an unknown amount.

And no causal comparison on its own. Two Kaplan–Meier curves for two groups are two descriptions. If the groups differ in anything else, the difference between the curves includes that too, and Simpson’s reversal applies here as everywhere: a treatment can appear better in every stratum and worse overall.

The first two are limits of the estimator. The third is a limit of any descriptive comparison and is worth saying because survival curves are unusually persuasive to look at — two separated curves with a gap between them read as a demonstration, and they are a description that still needs everything a causal claim normally needs.

What the field established #

Two essays, three measurements, and the pattern is one this site keeps producing.

Censoring is information, and the two obvious ways of handling it are wrong by thirteen and thirty-one percentage points against a known truth.

The estimator that uses it properly is exact to within a fraction of a point at every censoring level from 37% to 71%, and its stated uncertainty is honest — the estimate sits within four of its own standard errors of the truth at every time checked.

And the assumption it rests on cannot be checked from the data, so it has to be argued for from how the study was run rather than tested.

The last is the one worth carrying furthest. Every field on this site ends up at the same place: the arithmetic can be made exact and checkable, the properties of a procedure can be counted rather than claimed, and the thing that finally decides whether the answer is right is an assumption about how the data came to exist — which no amount of counting reaches.

How the estimator arrives at the right answer #

It is worth walking one step of the product explicitly, because the mechanism is more convincing than the formula and it explains why the naive readings fail where this succeeds.

Suppose at eighteen months there are 84 subjects still under observation and 3 have the event at that time. The estimated probability of surviving that instant, given survival up to it, is 1 − 3/84 = 0.964.

Now suppose 5 subjects are censored at eighteen months as well. They do not appear in the numerator — they had no event — and they do appear in the denominator, because they were being observed at the moment the three events occurred. After that time, the risk set drops by 8: the three events and the five censorings.

Both facts are correct and neither requires knowing what happened to the five afterwards.

Compare with the naive readings. Dropping them removes their contribution to the denominator at eighteen months, which inflates 3/84 into 3/79 and pushes the curve down — and the effect compounds across every event time. Treating them as events puts them in the numerator, making it 8/84 and pushing the curve down much harder.

So both errors have the same origin: misplacing the censored subject in the fraction. One removes it from a denominator where it belongs; the other adds it to a numerator where it does not. That the two errors run in the same direction is now obvious rather than coincidental — both make the fraction larger than it should be.

Two routes to the estimate #

The site’s discipline requires a second route, and for this estimator there is a clean one worth stating.

The Kaplan–Meier estimate is the non-parametric maximum likelihood estimate of the survival function under independent censoring. That is a derivation from a completely different direction: write down the likelihood of the observed data — a density term for each event, a survival term for each censoring — and maximise it over all possible survival functions. The maximiser is the product above.

So the estimator is not an ad hoc bookkeeping device that happens to work. It is what the likelihood prescribes, and the conditional-product construction is a way of computing it rather than the reason for it.

The empirical route confirms it independently. Run the estimator against data generated from a known exponential survival curve and average across fifteen hundred studies: the estimate lands within a fraction of a point of the true value, at every censoring level and at every time checked.

Three routes agreeing — the conditional product, the likelihood, and the simulation against a known truth — is the pattern this whole site is built on, and it is why the field’s headline result can be stated without hedging.

When a parametric model is better #

An honest counterweight, since the essay has been enthusiastic about a method that assumes nothing about the shape of the curve.

Kaplan–Meier is non-parametric, which means it makes no assumption about the survival distribution’s form. That is its strength and it has two costs.

It cannot extrapolate. A parametric model — exponential, Weibull, log-normal — can estimate survival at five years from three years of follow-up, because the assumed shape carries the estimate forward. Kaplan–Meier stops at the last observation. Whether the extrapolation is worth having depends entirely on whether the shape assumption is right, and it is usually unverifiable in the region that matters.

And it is less efficient when the shape is known. With a genuinely exponential process, fitting an exponential uses every observation to estimate one parameter and produces a much tighter estimate than the step function does. The non-parametric estimator is paying for its generality in precision.

So the choice is the familiar one on this site. The method that assumes less is safer and wider; the method that assumes more is sharper and depends on the assumption. The reasonable default is Kaplan–Meier for description within the observed range, and a parametric model only where extrapolation is genuinely required and the shape can be defended — with both reported, so a reader can see how much the assumption contributed.

Reading a published curve #

Four things a reader can check on any survival figure, none requiring the data.

The numbers at risk. Good practice puts a row beneath the axis giving how many subjects remain at each time point. It is the single most informative addition to the plot: it says where the curve is well determined and where it is being drawn from a handful of people. A curve with no risk table is hiding the part that decides how much of it to believe.

Where the censoring marks are. Ticks along the curve show censored subjects. Heavy censoring early is a different situation from heavy censoring at the end — the first suggests dropout and raises the independence question, the second is administrative and benign.

Whether the interval widens. It should, steeply at the right. A curve with a uniform band around it is not showing Greenwood’s error and is misrepresenting the late estimates.

And where the curve stops. If it ends above zero, the median or later quantiles may not be estimable, and any quoted mean is an extrapolation. A curve truncated at the point the risk set became small is honest; one drawn confidently to the end of the axis usually is not.

Those four take under a minute and they separate a figure that supports its conclusions from one that presents them.

The measurement to carry away #

Kaplan–Meier gives 0.499 against a truth of 0.497, at 71% censoring. Dropping the censored subjects gives 0.072 against the same truth.

Those two numbers come from the same dataset, analysed two ways, and the difference between them is larger than the entire quantity being estimated. No amount of care in the rest of the analysis — the design, the sample size, the significance testing, the correction for multiple comparisons — recovers anything from that choice going the wrong way.

Which is the ordering this site keeps arriving at from different subjects. Getting the structure of the data right dominates everything downstream of it, and it is usually the decision made fastest and documented least.

Why this estimator is on this site #

A closing note about fit, since survival analysis is a large field and two essays are a small sample of it.

Kaplan–Meier belongs here because it is a rare case of a method whose correctness can be demonstrated completely. The truth is available — the data is generated from a stated rule — the estimator can be run against it thousands of times, its bias can be counted, and its stated uncertainty can be checked against its actual error. Very few methods in applied statistics can be examined that thoroughly, and the ones that can are the ones worth using as reference points.

It also carries the site’s rule into new territory. Nothing is called 95% until it has been counted applies to a survival estimate exactly as it does to a proportion interval, and the version of the check used here — requiring the estimate to sit within four of its own standard errors of the truth — is the strongest form the rule takes anywhere on the site, because it tests the error bar and the estimate together.